Metabolites 2012, 2, 529-552; doi:10.3390/metabo2030529
OPEN ACCESS
metabolites
ISSN 2218-1989
www.mdpi.com/journal/metabolites
Review
Optimality Principles in the Regulation of Metabolic Networks
Jan Berkhout 1,2,? , Frank J. Bruggeman 1,3,4 and Bas Teusink 1,2,3
1
Systems Bioinformatics, AIMMS, VU University, 1081 HV, Amsterdam, The Netherlands
Kluyver Centre for Genomics of Industrial Fermentation/NCSB, 2600 GA, Delft, The Netherlands
3
Regulatory Networks Group, Netherlands Institute of Systems Biology, Amsterdam, The Netherlands
4
Life Sciences, Centre for Mathematics and Computer Science (CWI), 1098 XG Amsterdam, The
Netherlands
2
?
Author to whom correspondence should be addressed; E-Mail: [email protected],
Tel.: +31-(0)-20-5982705
Received: 20 July 2012; in revised form: 15 August 2012 / Accepted: 17 August 2012 /
Published: 29 August 2012
Abstract: One of the challenging tasks in systems biology is to understand how molecular
networks give rise to emergent functionality and whether universal design principles apply
to molecular networks. To achieve this, the biophysical, evolutionary and physiological
constraints that act on those networks need to be identified in addition to the characterisation
of the molecular components and interactions. Then, the cellular “task” of the network—its
function—should be identified. A network contributes to organismal fitness through its
function. The premise is that the same functions are often implemented in different
organisms by the same type of network; hence, the concept of design principles. In biology,
due to the strong forces of selective pressure and natural selection, network functions can
often be understood as the outcome of fitness optimisation. The hypothesis of fitness
optimisation to understand the design of a network has proven to be a powerful strategy.
Here, we outline the use of several optimisation principles applied to biological networks,
with an emphasis on metabolic regulatory networks. We discuss the different objective
functions and constraints that are considered and the kind of understanding that they provide.
Keywords: metabolic regulatory networks; optimal regulation; systems biology; design
principles
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1. Introduction
The availability of genome-wide datasets is increasing rapidly. Surprisingly, more data does not, per
se, lead to better understanding. In other words, a mechanistic understanding of biological networks that
are used to generate these datasets is lacking behind. One reason for being so is the emergent behaviour
that arises from the interactions within these networks. Consequently, the systemic consequences of
molecular perturbations cannot be predicted, nor does it lay bare the molecular mechanisms leading
to cellular behaviour. Systems biology develops methodology to overcome these limitations and
supplements molecular cell biology. Systems biology aims to understand the (dys-) function of living
organisms by studying the dynamics of molecular networks; effectively by answering how molecular
interactions give rise to cellular behaviour [1–3].
To achieve understanding of the function of a molecular network and how this results from molecular
interactions requires the identification of the molecular make-up as an obvious first (and important) step.
This often involves whole-genome sequencing, additional biochemical knowledge (“legacy data”) and
experiments. Such approaches have led to the determination of the topology of metabolic networks for
a great number of organisms [4,5]. Yet, how those molecular networks are rewired during adaptive
responses, i.e., how they are regulated and why, is less well understood [6]. This problem is so
challenging because it requires the integration of signalling, gene, and metabolic networks. Metabolic
networks alone are already composed of thousands of interactions between enzymes and metabolites.
The activity of metabolic networks results in a complex manner from internal constraints (such as
enzyme kinetics, physico-chemical parameters, thermodynamics) and the interplay between environment
and the impact of the associated signalling and gene network. Together these networks form, what we
will call, the metabolic regulatory network (MRN, Figure 1). This complex network displays multiple
overlapping regulatory interactions and feedback regulation. The recognition of this complexity has
inspired many studies to integrate datasets from different cellular regulatory levels [7–12]. These and
other studies often address how activities or components are affected by environments. Yet, a very
different question is why this particular behaviour is there in the first place. Is it possible to understand
the design principles of MRNs? We think we can, and we think that such understanding will help in
finding the generic biological structures underlying the piles of data that are being gathered with todays
technologies.
Figuring out the design principles of a MRN compares to a reverse engineering strategy. In order
to design a certain device, an engineer needs a system specification saying what the system should
do—what its function is—and all the relevant constraints should be considered, e.g. the production costs,
life-time of the device, etc. When one studies a biological design, the reverse approach is required. The
function—arising from the “specs”—and the relevant constraints need to be identified. When a similar
function occurs across organisms or networks and when those are achieved by similar networks, a design
principle is found. In other words, a design principle can be seen as the mechanism for network functions
that have proven to be successful in evolution. Such principles likely have emerged as the outcome of
similar selective pressures. Thus, in biology, it makes sense to identify design principles as the result of
an optimisation process of (a determinant of) fitness. The outcome of an optimisation of a given fitness
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measure and/or a given set of constraints is then taken as a hypothesis that can either be falsified or
verified [13]. This process will be discussed in this review (Figure 2).
Figure 1. Overview of regulatory interactions involved in metabolic regulatory networks.
The function of metabolic networks are governed by constraints. The regulation of a
metabolic network involves a tight interplay between different cellular networks such as
signalling and gene networks and by interactions with its environment. The enzyme
capacity is the net result of the amount of enzyme expressed and its activity as dictated
by post-translational modification and allosteric regulation. Metabolite pools and fluxes are
considered as the outputs of metabolic reaction networks and can be involved in various
regulatory feedback loops to other networks within the metabolic reaction networks as
indicated by the dashed arrows.
Metabolic regulatory network
signalling
network
covalent
modification
constraints
posttranslational
modification
environment
Metabolic
network
gene
network
gene
expression
allosteric
regulation
enzyme
capacity
Metabolites,
Fluxes
We thus start from the fact that adaptations are an inherent property of all living organisms, which is
to be explained by natural selection. However, that does not mean that the adaptations are always perfect,
nor that fitness is always optimal, only that fitness changes in a direction of increase. Thus, we try to
gain insight by studying specific examples of adaptation, in the light of natural selection and historical
interactions and constraints (see also [13,14] and a classical paper criticising optimisation approaches by
Gould and Lewontin [15]).
An example of one such optimality hypothesis is the economy of protein expression. The typical
large number of enzymes in metabolic networks results in a significant burden on total cellular resources
[16–19]. Cellular fitness can only increase if the right enzymes are expressed at the right time. Thus,
the regulation, or the lack thereof, of metabolic networks can have large beneficial or adverse effects
on cellular fitness, as shown experimentally [18,20,21]. Alternatively, biological systems have been
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exploited to achieve other control objectives. Examples in bacteria include: optimisation of growth rate
[22–24]; optimal swimming pattern [25,26]; adequate timing of transcription of amino acid metabolic
enzymes [27]. The above examples have in common that they all use an optimality principle to
understand systems behaviour. Some approaches, both theoretical and experimental, aim to yield
fundamental insights into the design principles of MRNs. Other approaches, such as flux balance
analysis (FBA) simply assume such design principles exist and explore optimality to predict behaviour
at a genome-scale. We discuss the relative advantages and limitations of these approaches, with a focus
on the type of optimality hypothesis that is relevant (Figure 3). Finally, some open questions regarding
MRNs are provided.
Figure 2. Schematic overview of the interactions involved in the process of evolutionary
optimization of metabolic regulatory networks. Constraints limit the functionality of a
metabolic reaction network (MRN), which for a given environmental condition can be
analysed with respect to (a) certain objective function(s), giving rise to some fitness.
Depending on selective pressures (which in turn are also dependent on the environment),
natural selection acts on the fitness of a metabolic reaction network.
kinetic (dynamics)
cost (protein burden)
stoichiometry
thermodynamics
....
}
environmental
sensing
MRN
constraints
optimal yield
robustness
growth-rate
....
}
selective
pressure
natural
selection
objective
function(s)
fitness
2. Optimal Control of Metabolic Reaction Networks for Semi-Autonomous Modules
Biological functions can rarely be attributed to single molecules; instead it is the result of many
interacting molecules within cells. A convenient way of studying biological function is therefore by the
analysis of biological modules [28,29]. As proposed by Hartwell et al. [30] “Modules are composed
of many types of molecules. They have discrete functions that arise from interactions among their
components,. . . , but these functions cannot easily be predicted by studying the properties of the isolated
components.” Inspired by the similarities between biological modules and man-made objects, analyses
from control and information theory has been applied to biological modules, yielding insight in the
general principles that govern the function of biological systems, including MRNs [12,26,29]. One can
think of a module as a sub-network inside the cell that can carry out some task nearly irrespectively
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of the state of the remainder of the molecular network inside the cell [31]. Examples of such systems
are stress-response systems, (e.g., the heat-shock system and osmotic shock response), iron scavenging
systems, or two-component signalling systems in bacteria. These systems function semi-autonomously
inside cells. Even though modules occur within molecular networks, not all functions derive from
them. For instance, current data does not suggest that metabolism can be perceived as a mosaic of
weakly coupled modules engaged in mass exchange: one responsible for alanine synthesis, another
for ATP generation, or for DNA synthesis. It is much more likely that metabolism operates as a unit
with strong interdependencies between subnetworks that carry out different functions. This should also
have consequences for its control and regulation by signalling and gene networks: i.e., about the design
principles of MRNs.
Figure 3. Different approaches to study metabolic regulatory networks classified according
to the level of detail. Depending on the level of detail of analysis, different objective
functions can be addressed. Pathway analysis refers to the analysis of (detailed) reactions
that might be embedded into a pathway. With this type of analysis why questions can be
addressed. Semi-autonomous modules function independently of the rest of the network,
and have a discrete function. Concepts from control and information theory have been
applied to understand how functionality emerges from the molecular components of these
modules. Genome scale models uses the information content from the entire genome to
figure out what flux distribution may lead to an optimal behaviour.
pathway analysis
optimality principles:
level of detail:
constraints:
characteristics:
examples:
cost of protein expression
(detailed) pathways; enzyme kinetics
protein burden; resource limitation
quantitative; studies ‘’why’’-questions
optimal protein expression; optimal
stoichiometry; growth strategies
Metabolic regulatory network
signalling
network
covalent
modification
constraints
posttranslational
modification
semi-autonomous modules
optimality principles:
level of detail:
constraints:
characteristics:
examples:
maintaining biological functions
modules
kinetic; dynamics; adaptations
qualitative; studies ‘’how’’-questions
perfect adaptation; chemotaxis
fold change detection
environment
Metabolic
network
gene
network
gene
expression
allosteric
regulation
genome scale models
enzyme
capacity
Metabolites,
Fluxes
optimality principles:
level of detail:
constraints:
characteristics:
examples:
optimal yield
entire pathways, genome-scale
stoichiometric; steady state
quantitative; underdetermined systems
studies ‘’what’’ -questions
FBA; rFBA; ROOM; MoMa
2.1. Maintaining Biological Function in Dynamic Environments
In their natural habitat nearly all organisms encounter environmental dynamics that require adaptation
in order to improve fitness. These observations can have different interpretations with respect to the
aim and consequences of the networks involved. Here, we will argue that although adaptations are
indispensable, they are not the objective. Rather, it is maintaining biological functions (in varying
environments), which is in the end achieved through adaptations [14].
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Robustness is a fundamental property of biological systems [32–36]; it allows a biological system
to perform its functions despite external or internal perturbations. This necessitates robust systems to
behave in a flexible way upon disturbances without affecting the performance/function of the perturbed
network. All changes should compensate for reductions in function upon perturbation. In other words,
robustness is a mechanism that allows changes in the structure and components of the network, while
specific functions of the network are maintained. While robustness is a key feature of biological systems,
they should at the same time be evolvable; e.g., large changes in fitness should be possible as a result of
only a few mutations. This is an intriguing aspect of biological systems; robust to specific perturbations
while evolvability is maintained. In addition, a system which is robust to all types of perturbation is thus
likely to lead to an evolutionary conservatism that inhibits the discovery of new adaptive solutions [37].
Cells often achieve robustness through feedback circuitry or the activity of a signalling network,
which sense environmental changes and induce compensatory or adaptive responses.
Many
characteristics of signalling modules have been studied by analysing their input-output relationship. A
feature displayed by several signalling modules is perfect adaptation. Arguably the best studied module
for perfect adaptation is bacterial chemotaxis in Escherichia coli. The signalling network in bacteria
responsible for chemotaxis has served as a classical example for understanding how functions at the
network level emerge from interactions between constituent molecular components. In this network,
as shown in Figure 4A, changes in attractant or repellent concentrations (purple star) are sensed by a
protein complex consisting of transmembrane receptors, adaptor protein CheW (blue), and a histidine
kinase CheA (orange). Autophosphorylation activity of CheA is inhibited by attractant binding and
enhanced by repellent binding to receptors. The phosphoryl group is transferred from CheA to the
response regulator CheY (yellow). The level of phosphorylated CheY (Y-P) influences the direction of
the flagellar motors. Adaptation is mediated by the action of two enzymes: CheR (green) and CheB
(brown), which adds or removes methyl groups on each receptor monomer. Feedback is provided by
CheB phosphorylation through CheA that increases CheB activity. Chemotaxis allows bacterial cells
to swim in the direction of favoured chemical attractants (such as food) or away from repellents (such
as toxic compounds) by changing the frequency of tumbling [38,39] (for reviews on chemotaxis see
[40,41] and additional references therein). Interestingly, the mechanism for chemotaxis is based on a
robustness feature called perfect adaptation (Figure 4A) [25,26,35,36]. The steady-state behaviour of the
system is independent of the concentration of the attractant or repellent, which is achieved by resetting
the output value to the pre-stimulus level. Barkai and Leibler showed, using computer simulations,
that perfect adaptation is not dependent on the fine-tuning of parameters in the network; instead, it was
shown that perfect adaptation was an intrinsic property of the connectivity of the signalling network [35].
Experimental evidence for this robustness property in chemotaxis was provided by Alon et al. [36]. In
an elegant experimental set-up these authors showed that perfect adaptation was not affected when the
levels of chemotactic proteins were changed over a wide range. This inspired other studies to apply
concepts from control and information theory resulting in the observation that an integral control system
is key for robust perfect adaptation [26,42,43]. This becomes evident when the molecular network is
represented as a block diagram (Figure 4A), with the chemoattractant as input and receptor activity
(Y-P) as output. −x represents the methylation state of the receptors. The difference between output (y1 )
and a reference value (y0 ) represents the error. Integral control arises through the feedback loop in which
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the integrated error is fed back into the system, leading to perfect adaptation. A recent computational
search of all possible three-node networks which are able to perform perfect adaptation confirmed this
observation [44].
Perfect adaptation was also experimentally demonstrated in the regulation of osmotic shock in yeast
[45]. By means of single-cell analysis of the dynamic response of the hyper osmotic shock network, these
authors demonstrated that nuclear enrichment of a MAP kinase (Hog1) displays perfect adaptation. The
requirement of an integral feedback system to observe perfect adaptation was exploited to reveal the
importance of Hog1 kinase activity.
In contrast to perfect adaptation, another observed adaptive response is fold change detection (FCD)
[43,49,50]. In FCD the dynamics of the output is only dependent on the fold-change in the level of
the input signal, and thus not on the absolute levels of the signal. A system that displays FCD will
produce identical outputs (not only the magnitude of the response peak but also the entire profile over
time depends only on the change ratio of the stimuli) in response to, for instance, a change in input signal
from 2 to 4 compared to 10 to 20 (Figure 4B). Modules that display FCD should thus display some sort
of memory in order to compare the change in input signal to the previous state. Recently, it was shown in
a theoretical approach that FCD can be generated by a so-called incoherent feedforward loop (I1-FFL)
[49]. In a I1-FFL, an activator, X, activates a repressor Y, that in turn control target gene Z, which is also
directly activated by X. Other examples of network motifs that can display FCD are a nonlinear feedback
loop and a linear integral feedback system [51].
What are the advantages of FCD? It is postulated that having a FCD (i) ensures an adequate response
towards an activator that is known to naturally vary up to many folds and (ii) the ability to maintain
sensitive despite noise in the input signal [49] (see also section 2.2). The latter becomes clear when
considering two inputs: a low signal and a high signal. If noise is in some manner proportional
to the level of signal, a high level of noise is unavoidable for the high signal, which might cause
fundamental problems for absolute detection mechanisms. In other words, FCD allows organisms
to respond optimally to a gradient and this response is invariant of multiplying the gradient with a
constant [51].
Another characteristic of a l1-FFL motif is its ability to generate a pulse and accelerate the response.
This behaviour is the result of the two opposing inputs (X activates Z and Y; Y represses Z) that are
integrated by Z. Upon an increase in X, Z is rapidly increased, but after some time, Y builds up and
starts to decrease Z again. As a result, initially Z rises rapidly, and then its concentration drops again,
leading to pulse-like dynamics. An example of this motif can be found in the gal operon in E. coli:
external galactose is transported into the cell by a permease (GalP). Internal galactose (gal) binds to
gal isorepressor (GalS), which negatively regulates the gal operon. External glucose inhibits production
of cAMP which, when bound to protein Crp, acts as an activator of the gal operon. I1-FFL speeds
up a response (indicated by the red dots) and generates a pulse upon stimulation compared to simple
regulation (Figure 4B). The acceleration of the response, up to 3-fold compared to a simple regulation
system, arises when GalS does not completely inhibit galETK. Disruption of the l1-FFL motif, by
mutations or artificial conditions, abolished these dynamics [52].
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Figure 4. Illustration of network motifs to study molecular networks. In the left column
the molecular interactions underlying the networks are shown, the middle column shows
the network motif and corresponding characteristics are plotted in the right column. The
examples shown here, are discussed in detail in the main text. A Chemotaxis signalling
network in E. coli. (Figures adapted from [26,46]). B Catabolite repression of the gal operon
in E. coli. (Figures adapted from [31]). C Competence in Bacillus subtilllis. (Figures adapted
from [47,48]).
network motif
characteristics
chemotaxis in E. coli
integral feedback control
- perfect adaptation
z
CCW
catabolite repression
of gal operon E. coli
system response
input-function
Y
time
∫
I1-FFL network motif
- response acceleration
- pulse generator
cAMP
glucose
galactose
GalP
Crp
cAMP
gal
GalS
I1-FFl
galactose
simple
regulation
time
GalS
- fold change detection
input
galETK
galETK
time
vegetative
competence
positive feedback loop
quorom sensing
ComK proteolytic
degradation
sporulation
based on single cell measurements
- passing the threshold
triggers competence
- bistability
ComK
treshold for
autostimulation
autostimulation
ComK
competence of B. subtilis
time
ComK
C
identical
output
ComK
monodal
ComK
time
ComK
bimodal
transcriptional
control
ComK
regulon
Rok
ComK
time
non-competent
time
competent
ComK
Crp
+
time
-x
CW
B
y1
k
response
+
output
B p
Yp
Y
A
input
w
y0
R
output
A
molecular network
time
2.2. Managing and Profiting from Inevitable Molecular “Noise”
The molecules that make up biological networks are continuously synthesised and degraded, where
the latter process results from intrinsic degradation and/or via dilution due to cell growth. Synthesis and
degradation rates should be balanced to maintain a molecular species at a constant level over time. Due
to the inevitable asynchrony in the individual synthesis and degradation events, fluctuations in molecule
numbers are such that transient deviations from the steady state level occur. Those fluctuations occur
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in single cells and remain invisible when populations of cells are studied. When single cells were
studied it became clear that molecular species, which occur in small numbers, such as transcription
factors, can display large fluctuations—“noise”; the coefficient of variation (defined as the ratio of the
standard deviation to the mean) ranges in 25%–40%. Even though such fluctuations are transient, they
do cause alterations in network function because they induce variable protein expression levels. In
this manner, fluctuations in molecular species can lead to phenotypic differences between genetically
identical cells living in a homogeneous environment [53,54]. In depth discussions can be found in
reviews about experimental approaches to noise in gene expression [55,56] or about the mechanisms
generating population-level variability [57–59]. Here, we will focus on how cells cope with molecular
noise, as sometimes they aim to minimise it, while in other cases they profit from it. Each of
these cases requires particular network topological features and resulting dynamic behaviours. A few
design principles have been discovered that indicate quite universal strategies for how cells cope and
exploit noise.
In many cases, molecular regulatory networks are inevitably distorted by noise—although it remains
to be shown how much of a reduction in fitness it actually causes. One situation where noise can cause
a reduction in cellular performance is when it occurs in signalling networks. Such systems respond
to environmental changes and show changes in their output, e.g., a transcription factor, upon a change
in external signal level. But, if the components of the signalling network spontaneously fluctuate in
their level and their activation state, these systems can show changes in their output even at a constant
signal level: this is when they make an error. This effectively reduces the capacity of cells to distinguish
alternative environmental states. The number of distinguishable states can be related to the characteristic
size of the change in the output level due to changes in the signal level divided by the size of the output
change induced by spontaneous fluctuations at a fixed signal level. This measure thus compares the level
of desired signal to the level of background noise and is called the signal-to-noise ratio; a central concept
in information theory [60]. Recently, Cheong et al. [61] determined the number of environmental states
that can be distinguished by the NFκB signalling network in mammalian states and found a surprisingly
low number of only two states. Estimates by Tkacik and Bialek [60] show similar numbers for other
systems, indicating that signalling networks are severely reduced in function by noise and that cells
function despite this low performance.
Cells can also profit from noise, particularly when they are confronted with unpredictable
environments where sudden stress or food shortages can occur. One well-studied example is
the sporulation and competence response of Bacillus subtilis upon sudden stress conditions [47].
Competence is a physiological state, distinct from sporulation and vegetative growth, that enables
cells to bind and internalize transforming DNA. Based on single cell measurements, cells have either
a single (low or high) unimodal or bimodal distribution of some indicator for competence (distributions
shown as an example only, Figure 4C). Regulation of the activity of ComK, a transcription factor in the
competence network, is controlled through proteolytic degradation, quorum sensing and transcriptional
control. ComK binds to its own promoter and is required for its own expression. The competence
regulatory network therefore contains a positive feedback loop. Due to the positive feedback loop ComK
proteins (red circles) auto-stimulates itself (green arrow) after passing a threshold (green line) leading
to competence. When the ComK concentration does not cross the threshold ComK levels remain low
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and cells stay in a non-competent state as indicated on the left side. Since the start, duration, and size
of the stress is unclear and causes cell death, cells should prepare themselves and make sure that some
of them survive—they all have the same DNA so altruistic behaviour can actually occur. In growing
populations, a certain fraction of cells expresses the key regulator of competence ComK at high levels
and acquire competence for DNA uptake while the majority of cells remain in the low ComK expression
state. In this manner, a fraction of the population is prepared for the stress while others continue to
produce offspring as long as the conditions allow for this and in this process make some more dormant
cells. The exact probability for a cell to become dormant or proliferative is not fixed and is actually under
cellular control. Whether a given cell becomes proliferative or dormant is a stochastic event achieved
by molecular noise. This does not only occur in B. subtilis but also in other cells. The design principle
here is that a bistable network is used that can switch between states due to spontaneous fluctuations
in the levels of its molecular components (e.g., ComK expression level, Figure 4C), i.e., stochastic
phenotype switching.
3. Global Pathway Analysis
3.1. Optimal Gene Expression in Un-branched Metabolic Pathways
The group of Heinrich pioneered the application of optimality principles to biological systems
[62]. They studied the control structure of metabolic systems at states of optimal activity [63,64], the
optimal timing of metabolic gene expression [65], optimal gene expression patterns were found under
the assumption that expression patterns serve as regulators of cell functions [66], and tested whether
metabolic network design and stoichiometry could be the outcome of an evolutionary optimisation
process [65,67]. The optimal timing has been illustrated experimentally, whilst the other theories have
inspired new ways of thinking about biological systems but await experimental proof—provided such
experiments indeed can be done.
In the aforementioned experimental study, Zaslaver et al. measured the promoter activity of
amino-acid biosynthetic genes in E. coli over time, using GFP and Lux libraries [27]. Inspired by
earlier work from Heinrich [65], these authors used a mathematical analysis to unravel the underlying
objective of the observed timing of gene expression (a.k.a. just-in-time transcription). Based on their
analysis, it was concluded that just-in-time transcription is beneficial when selective pressures act on
rapidly reaching a new steady state with minimal enzyme production costs [27].
3.2. Playing the Optimality Game
Optimisation of organismal fitness is not always as straightforward as discussed so far. This has to
do with the fact that optimal biological traits or designs should also be evolutionary stable. Excretion of
extracellular proteins is such an intriguing trait. Let us take the excretion of extracellular protease by the
lactic acid bacterium, Lactococcus lactis, as an example. This protease is involved in the degradation
of milk proteins to form free utilisable peptides, as is needed to support growth, in conditions where
they are not (or not enough) supplied directly into the medium. One might predict that, under such
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conditions, the trait of excreting the protease will be selected for; after all, excretion of the protease
lead to more free peptides that are needed for cellular growth. However, since the proteases are excreted
extracellularly, the peptides produced do not merely benefit the cell secreting the protease, but in part also
diffuse away from it, becoming accessible to neighbouring cells. Imagine the scenario where one cell
does not excrete the protease. This “cheater” cell does not have the burden of production and excretion,
but is still able to take up free peptides. On the long term, this cheater can thus, on average, generate
more offspring leading to an increase in the protease-negative trait. Ultimately, the trait can completely
disappear in a population, as was shown experimentally [68]. Similar results were obtained in yeast that
excretes invertase extracellularly. The invertase is involved in the breakdown of sucrose into glucose and
fructose. The latter two are able to diffuse away from the cells that excreted the invertase, and therefore
accessible to other cells [69]. These two examples are a very counterintuitive outcome of the effect of
selection on the physiology of a species, even under constant conditions. A detailed theoretical analysis
of this cooperative and cheating behaviour was reviewed recently [70]. In conclusion, when fitness of a
phenotype is dependent on its frequency relative to other phenotypes in a given population, game theory
approaches are useful tools because they take evolutionary stability of traits into account.
3.3. Growth Rate Optimisation Shapes Growth Strategies
Cellular behaviour is governed by interacting networks. How is it possible to explain a global cellular
property, such as the growth rate, if one studies only a sub-part of the network? This question inspired
Molenaar et al. to propose a self-replicating model of a cell that is composed of a minimal set of required
modules, such as a module for the ribosomes, metabolic pathways, substrate transporters and lipid
biosynthesis [71]. The essence of this self-replicating model is: for a given environment, total cellular
resources are optimally allocated by the ribosomes (including allocation to synthesis of new ribosomes),
such that the growth rate is maximised. Using this approach, it was shown how the experimentally
frequently observed shift between metabolically or energy efficient and inefficient metabolism can
be explained as the result of optimising the cellular economy for growth rate. Other optimisation
approaches, including FBA (see section 4.1) and game theory (see section 3.2), have failed to explain
such shifts in metabolic strategies because they will always select one of the strategies irrespective of the
external substrate level.
3.4. Optimal Protein Expression Levels Maximise Growth Rate
Experiments in which enzyme levels were titrated using inducible promoters provide indirect but
compelling evidence that microorganisms optimally tune enzyme levels [72–74]. Interestingly, these
experiments showed that the growth rate of E. coli is maximal at the wild type level of ATPase expression,
indicating that in E. coli—for the enzymes considered—protein levels are fine-tuned to result in the
highest growth rate. Similar results have been described for glycolytic enzymes in L. lactis [75,76] as
well as the activity of the las operon [77]. In addition, laboratory evolutionary experiments [78], by
means of serial propagation [79], also confirmed that protein expression levels optimise cellular fitness.
However, the resources a microorganism has at its disposal are limited. This implies that expressing
and maintaining enzymes is costly. Indeed, it has been observed that expressing unneeded proteins
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has a negative effect on the growth rate [17–19]. Depending on the environmental conditions, protein
expression can also be beneficial. Therefore, in order to optimise fitness, a precise balance between these
two processes is required.
Above examples show that mutations affecting the amount of protein expressed are an adaptive
mechanism to deal with changing environments and thus a target for selective pressure to act on.
However, these observations do not provide an explanation in terms of the underlying design principle.
This was provided by Dekel and Alon, who proposed a quantitative model in terms of the underlying
costs and benefit of protein expression [20]. Using the lac operon in E. coli, these authors were
able to experimentally measure the cost and benefit of protein expression of β-galactosidase (LacZ),
which hydrolyses lactose into glucose. While increasing the expression level of this enzyme will
increase the cost, the benefit of its activity will increase with increasing lactose concentrations. Cost
and benefit were measured by the effect of different protein expression levels on the growth rate at
different lactose concentrations. The optimal expression level—that is, where the difference between
benefit and cost is maximal—was measured for different lactose concentrations. Interestingly, a serial
propagation evolution experiment, at different lactose concentrations, revealed that cells adapt their
protein expression level to the predicted optimal level (within a few hundred generations) [20]. This
indicates that, under the right environmental conditions, the selection pressure on a single protein can be
very strong.
Additionally, Eames and Kortemme recently presented an approach to perturb the production,
function and folding efficiency in redesigned E. coli strains, and showed that the lac permease (LacY)
activity is a major physiological source of expression costs in the lac operon [21]. The cost scales linearly
with LacY activity (and not with LacZ) and outweighs cost-effects of protein production and misfolding
in the lac operon. In light of the regulatory network of the operon, these results signify mechanisms that
minimise the physiological costs of LacY activity, such as the direct inactivation of LacY function in the
presence of glucose and other carbon sources, known as inducer exclusion.
The concept of a cost-benefit optimisation has also been applied to explain gene regulation functions
(the relation between input signals and the gene expression levels) of the lac operon [80]. Starting with
a measured gene regulation function, and assuming that this function has evolved to optimally suit the
natural environment of E. coli, the authors were able to explain the shape of the gene regulation function
in terms of underlying regulatory mechanisms. For example, the steep shape of the regulation function
at intermediate lactose levels is suggested to optimally minimise the effects of noise in an environment
with a bimodal distribution of lactose concentrations.
The definitions for cost and benefit as initially coined by Dekel and Alon [20] were explicitly given
in terms of the lac operon kinetics for E. coli. In order to make such an analysis applicable to metabolic
pathways, we recently presented generalised definitions for cost and benefit (Berkhout et al., submitted).
We propose quantitative and intuitive concepts for cost and benefit of enzymes in a metabolic pathway,
and calculate these from kinetic and biochemical data, under the assumption that fitness increases with
pathway flux. To test our predictions, a laboratory evolution experiment with Saccharomyces cerevisiae,
propagated on galactose, under growth rate selection, was performed. Interestingly, the experimental
data agree with the model predictions by showing that a high increase of growth rate is attained by
increasing phosphoglucomutase levels, as observed previously as well [81,82]. The above examples
Metabolites 2012, 2
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show that a cost-benefit analysis can shed light on biological principles such as protein expression and
gene regulation and provide a framework that carries strong predictive power.
3.5. Feasibility Analysis
The regulation of metabolic networks is the net result of the interplay between modulation of enzyme
levels [83] (sometimes called hierarchical regulation [7,84]), and metabolic regulation, e.g., via allosteric
interactions. Quantitative analyses that relate changes in metabolic fluxes to changes in transcript or
protein levels have revealed a remarkable lack of understanding of the regulation of metabolic networks.
Combining the why and how questions might be a step towards obtaining a better understanding of the
regulation of MRNs. Recently we have proposed such a method: feasibility analysis [85].
Feasibility analysis starts with a kinetic model of metabolism, of which the parameters are sampled
to create solution spaces (which are bounded by constraints) and a (sub-) set of these samples is selected
according to an objective function or multiple objective functions. Thus, feasibility analysis allows for (a
visual) discrimination between different modes of metabolic regulation, and evaluates conditions where
multiple objective functions have to be traded-off by cells. Using experimental data from long-term
carbon limited chemostat cultivation of yeast cells [86], with feasibility analysis, we were able to
understand the observed adaptations. We concluded that yeast cells, after long term adaptation, employ
a mixed strategy between (at least) two opposing strategies: decreasing enzymes expression levels as a
way to decrease enzyme overcapacity and investing in expression levels involved in taking up the limiting
growth substrate. This trade-off renders the cells specialised in a low-carbon flux state to compete for
the available glucose and get rid of the enzyme overcapacity.
Based on 13 C-flux experiments in different bacteria, it was recently shown that metabolism indeed
operates close to the optimal surface defined by multiple objective functions: maximum ATP yield,
maximum biomass yield and minimal sum of absolute fluxes [87]. Using flux data from evolved E. coli
on alternating carbon sources, it was proposed that the location in the flux state space (the feasible space
constrained by the three objective functions) can be understood from an underlying trade-off between
(near-) optimality under a given condition and minimal adjustment to alternative conditions.
An advantage of feasibility analysis is that the analysis is based on a kinetic model, therefore a more
extensive set of objective functions can be implemented. An example is the objective functions that
involve dynamics, which cannot be studied within FBA. Simultaneously, it limits feasibility analysis, due
to the number of kinetic models available. Although a kinetic model of the entire metabolic network of
an organism will be extremely useful to characterise fully the mechanics of each enzymatic reaction, and
to combine such knowledge to ultimately predict system-behaviour, we are far from that. Besides some
first attempts towards this goal [88,89], the number of complete kinetics models is sparse. Furthermore,
kinetic enzyme assays are often done under in vitro conditions which do not always resemble in vivo
environments [90–92]. For a depository of available and curated models of biological systems the reader
is referred to online databases such as JWS [93] and Biomodels [94]. Note that the former also allows
for the user to run these models online.
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4. Genome Scale Models
The arrival of whole-genome sequencing techniques has provided a comprehensive description of
the genetic content (the “parts-list”) that makes up an organism. The reconstruction of all (metabolic)
reactions within an organism requires the identification of all enzymes encoded by the genome and the
chemical transitions that they participate in. A complete description of the entire genome was long
thought to be the requirement to understand biological function (and ultimately to cure diseases). It
turned out, however, that due to elaborate regulatory networks these promises could only be partially
achieved. Nevertheless, these studies have facilitated the acceleration of the development of other
experimental techniques (for instance ChIP- and RNAseq) as well as the development of an active field
of research that develops genome-scale in silico models. These models are used to explore all flux
distributions within the network which results in optimisation of an objective function.
4.1. Flux Balance Analysis
Flux balance analysis (FBA), inspired by the premise that during evolution natural selection prompts
biological systems towards optimality, assumes that cells perform optimally with respect to a given
objective function. The optimisation of such an objective function is used to find a (sub-) set of optimal
states from the large solution space that is defined by the constraints. The optimisation of the production
of biomass or ATP are examples of commonly used objective functions.
Does the optimality assumption within FBA help in understanding biological systems? Yes and no.
Let us take the optimisation of specific growth rate (that is equivalent to maximisation of the rate of
production of biomass per unit biomass [95]) as an example. There is considerable evidence that,
especially under laboratory evolution experiments, the growth rate is representative of cellular fitness.
So, how does FBA optimise the growth rate? FBA relates (measured) input and output rates via the
equation:
substrate
µ = Ybiomass
· Vsubstrate
(1)
substrate
is the biomass yield with respect to the
where µ is the specific growth rate (unit h−1 ), Ybiomass
substrate (unit gram dry weight per mmol substrate, gDW mmol−1 ) and Vsubstrate is the specific
uptake rate of the growth substrate (unit mmol h−1 gDW−1 ). With the uptake rate fixed as an
environment-dependent capacity constraint, in order to find the highest µ, FBA just finds the flux
distribution with the highest yield. Thus, it should be realised that the maximisation of growth rate
substrate
is essentially done by finding a flux distribution that maximises Ybiomass
(see also [96]). Hence, FBA
can rather accurately predict flux distributions for cases where high-yield strategies are favourable [97];
however predictions using FBA fail when organisms display strategies whereby substrate is “wasted”
into byproducts, thereby lowering the biomass yield (high rate strategies). Indeed, E. coli [23] as well as
the lactic acid bacterium Lactobacillus plantarum [96] achieve optimal in silico predicted growth when
adapted on the poor substrate glycerol by serial dilution. However, on glucose, FBA predicted for L.
plantarum, biomass yields, which were too high and incompatible with the observed lactate production
[98]; similar discrepancies between model and experiment were observed for E. coli on glucose [23].
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Due to the still growing number of genome-scale models across all kingdoms, FBA is a powerful tool
to explore systemic properties of genome-scale metabolic networks for applications in biotechnology
and medicine [5,99]. Nevertheless, caution should be taken with (i) the selection of objective function
and (ii) non-uniqueness of the flux distributions that lead to optimisation of the objective function (i.e., a
whole solution space of flux distributions that is consistent with the prediction of the objective function).
In addition, point (ii) can be addressed by, for instance, flux variability analysis (FVA) [100]. FVA is
used to find the minimum and maximum flux values for the reactions in the network while maintaining
some state of the network (e.g., supporting 90% of the objective function).
4.2. Optimising the Predictive Power of FBA
The above examples point to an important aspect that obviously applies not only to FBA alone but
also to all analyses that assume some kind of optimality criteria: the predictive power critically depends
on the correspondence between the chosen objective function with the real biological objective. Indeed,
it was shown for E. coli that, within the FBA formalism, different objective functions were needed to
predict optimal flux states for different conditions [101]. The remaining part of this section will be
used to discuss different extensions and constraints of FBA that are proposed to unravel the principles
underlying the MRN.
The surmise that protein expression and maintenance is a costly process has also penetrated the field
of FBA. One way of implementing this is the minimisation of overall fluxes as the objective function.
The rationale for this approach is the assumption that the magnitude of fluxes is related to the amount
of protein required as catalyst of that flux. When applied to metabolic schemes of erythrocytes and
Methylobacterium extorquens the fluxes predicted by the method were in good agreement with those
calculated on the basis of a kinetic model [102,103]. Unfortunately, this objective too cannot explain why
microorganisms switch between metabolic strategies due to the fact that the flux-minimisation algorithm
will force some of the fluxes to zero if alternative “cheaper” reactions or pathways exist in the network.
The optimality assumption underlying FBA may be valid for wild type organisms that have been
evolved over many thousands of generations. It is less likely that this will also hold for engineered or
mutant-strains. To overcome this, extensions of FBA that use different objective functions are proposed,
such as ROOM [104] and MoMa [105]. Regulatory On/Off Minimisation (ROOM) aims to minimise the
number of significant flux changes with respect to the wild type. This method is based on the assumption
that upon a gene-knockout, the cost associated with adapting is minimised. In this manner, ROOM can be
used to predict the metabolic state of an organism after a gene-knockout. Another variant of FBA called
MoMA, which refers to the Minimisation of Metabolic Adjustment, seeks an approximate solution for
a sub-optimal growth flux state, which is nearest in flux distribution to the unperturbed wild type. Both
approaches have been shown to outperform FBA for certain specific examples [104,105]. Note, however,
that both methods require a reference flux distribution from the wild type to predict fluxes in the mutant.
In order to further improve the predictive power of FBA, there are options that do not act on the
objective function, but on the constraints that limit the ways in which the objective can be reached. An
example of an additional constraint is a molecular-crowding constraint. The rationale for this constraint
comes from the limited amount of space inside cells available for metabolic enzymes [106]. As argued
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544
in Section 3.3, FBA fails to predict a metabolic switch (changing from high-yield to low-yield metabolic
pathways with increasing substrate concentration). Recently, FBA extended with a molecular crowding
constraint was used in a comparative study between three organisms (E. coli, S. cerevisiae and L. lactis)
which differ in their metabolic strategies. With this extension, FBA was able to predict experimental
findings and it was reported that different metabolic strategies could be attributed to the design principle
of maintaining a redox balance within these organisms [107].
In the FBA formalism, constraints always act on the potential fluxes through reactions. First, one
tries to include additional data that can be used to constrain the fluxes, particularly for cases where
regulatory effects have a dominant influence on the behaviour of the organism’s metabolic fluxes.
Covert et al. proposed an extension to FBA using regulatory information (regulatory FBA, rFBA)
[108]. This was incorporated by means of a Boolean formalism, in which the state of a gene was
represented as either transcribed or not transcribed in response to regulatory signals. Based on whether
or not a gene was transcribed, the reactions catalysed by the corresponding enzymes were included or
removed from the stoichiometric model. Although a good start to include regulatory interactions, the
regulatory network was incomplete and imprecise. A better prediction can therefore be made when gene
expression data is used to infer regulatory networks [109]. This approach was also successful to indicate
knowledge gaps and identify previously unknown components and interactions in the regulatory and
metabolic networks [110].
There have been a number of other approaches to augment regular FBA with regulatory constraints.
Shlomi et al. [111] proposed steady-state regulatory FBA (SR-FBA). In addition to metabolic
constraints, in SR-FBA there are four additional constraints defining the type of regulation. The
combined functional state of the entire system in a given constant environment, referred to as
metabolic-regulatory steady state (MRS), is described by a pair of consistent metabolic and regulatory
steady states, which satisfy both the metabolic and regulatory constraints. Using the SR-FBA method,
the authors were able to show that a considerable number of 36 genes are redundantly expressed, that is,
they are expressed even though the fluxes of their associated reactions are zero. These 36 non-optimal
fluxes could not have been identified with a classical FBA approach and reveals that cells maintain
some sort of metabolic variability (or anticipatory behaviour) within a given growth medium. In another
approach, Van Berlo et al. suggested applying changes in gene expression as soft constraints on changes
in fluxes, i.e., it is to be expected that if the expression of a gene is increased, its corresponding flux
will also increase [112]. In their approach, flux distributions are sought that minimise the number
of violations to this rule and it was shown that changes in gene expression are predictive for changes
in fluxes.
Other examples, originating from biophysical limitations, that are used as additional constraints are
competition for membrane space [113] and grouping constraints based on genome context [114]. In the
first study it is hypothesised that the outcome of a competition for membrane space between glucose
transporters and respiratory chain proteins influences the ratio of respiration and fermentation. By
incorporating a sole constraint based on this concept in the genome-scale metabolic model of E. coli, they
were able to simulate respiro-fermentation [113]. The second study used three types of genomic context
(conserved genomic neighbourhood, gene fusion events, and co-occurrence) to estimate the likelihood
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of an entire group of fluxes to be on or off. Predictions of a genome scale model were in good agreement
with experimental data from E. coli [114].
FBA would greatly benefit from kinetic information of metabolic reactions. For some subnetworks,
such as glycolysis, these kinetic models are available, and they have been integrated with FBA (iFBA)
[115]. Focusing on the diauxic shift in E. coli the authors found a significant improvement over the
individual rFBA and kinetic model. Such approaches have the potential to combine the best of two
worlds: generating simulations which are more globally accurate and informative than the kinetic-based
model, and more accurate in their details than the rFBA model alone.
All approaches mentioned above, although coherent, still require further validation by datasets of
steady-state flux distributions and microarray data, and so at this stage it is difficult to assess which of
these methods is to be preferred. Finally, we would like to stress that predictions, true or false, are useful
either way because these predictions are testable and therefore able to confirm or rule out hypotheses.
5. Conclusion and Outlook
The overwhelming speed at which experimental data is currently generated has led to the recognition
that the complexity of the underlying networks is enormous. In order to make sense of this data, systems
biology is a scientific tool that can help in extracting design principles from it. In this review we have
presented experimental and theoretical examples of this extraction procedure based on the premise that
evolution has moulded biological networks to perform optimally. This assumption immediately raises
the question: “optimised for what?”. This is an intriguing and difficult question to answer. Optimisation
of an objective function is strongly related to cellular fitness (i.e., the ability to generate offspring),
but generally the link between fitness and molecular network properties is rather vague or indirect.
However, for several networks, especially under the standard laboratory conditions, we have quite a
good understanding of which underlying principles and mechanisms are responsible for optimising
fitness; many examples have been discussed in this review. For other networks it remains to be shown
whether the lack of understanding is due to (i) an incomplete topology of the network (e.g., missing
regulatory interactions), (ii) a lack in experimental data, or (iii) finding the right objective functions
and constraints. As argued throughout, we believe that focusing on points (i) and (iii) will shed most
light on understanding biological behaviour. Furthermore, care should be taken that biological networks
cannot be optimal at multiple objective functions at the same time. Multiple objectives have then to be
traded-off, which can be studied by means of a Pareto front [87]. In our opinion, a better understanding
of biological systems relies mainly on the identification and integration of objective functions and
constraints that have shaped current biological networks.
Acknowledgments
We thank Johan van Heerden for feedback on the manuscript. This work was supported
by Netherlands Organization for Scientific Research (NWO) Computational Life Sciences Project
(CLS-635.100.18). This work was part of the Kluyver Centre for Genomics of Industrial Fermentation
Metabolites 2012, 2
546
which is financially supported by the Netherlands Genomics Initiative. F.J.B. thanks the NISB, the CWI,
and the Dutch Organization for Scientific Research (NWO; Grant number 837.000.001) for funding.
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